Chapter 7.2, Problem 4CFU

To determine

To create: a new trigonometric identity for the given situation, explain the steps of creation and verify it.

Answer to Problem 4CFU

  $\tan x\cos x=\sin x$

The above trigonometric identity is created by using quotient identity.

This identity is verified by transforming the left side of equation into the right side.

Explanation of Solution

Given information:

The trigonometric identity must have three different functions.

Calculation:

The trigonometric identity having three different functions can be created by using quotient identity of tangent as shown below.

  $\begin{array}{c}\tan x=\frac{\sin x}{\cos x}\left[\text{Quotient identity}\right]\\ \tan x\cos x=\sin x\left[\text{Multiply both sides by }\cos x\right]\end{array}$

Thus the new identity having three different trigonometric functions is $\tan x\cos x=\sin x$ .

Now verify the identity by changing left side of equation into right side as follows:

  $\begin{array}{c}\tan x\cos x=\sin x\\ \frac{\sin x}{\cos x}\cos x=\sin x\left[\text{Quotient identity of tangent}\right]\\ \sin x=\sin x\end{array}$

Since left side of equation is transformed into right hand side. Thus it is verified that $\tan x\cos x=\sin x$ is an identity.